WEAK TYPE ESTIMATES FOR A SINGULAR CONVOLUTION … · fa(z, t) dzdPz dt = 0 and \a\ < \Q\~xXq (We denote by XB the character-istic function of a set 5 .) Let L(z) be a homogeneous

$Page 1: WEAK TYPE ESTIMATES FOR A SINGULAR CONVOLUTION … · fa(z, t) dzdPz dt = 0 and \a\ < \Q\~xXq (We denote by XB the character-istic function of a set 5 .) Let L(z) be a homogeneous$
TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 325, Number 1, May 1991

WEAK TYPE ESTIMATESFOR A SINGULAR CONVOLUTION OPERATOR

ON THE HEISENBERG GROUP

LOUKAS GRAFAKOS

Abstract. On the Heisenberg group H" with coordinates (z, t) e C" x R,

define the distribution K{z, t) = L(z)S(t), where L(z) is a homogeneous

distribution on C" of degree -In, smooth away from the origin and 5(t)is the Dirac mass in the t variable. We prove that the operator given by

convolution with K maps Hl(Wn) to weak l'(H").

1. Introduction

H" is the Lie group with underlying manifold C'xl and multiplication

(z,t)-(z ,t') = (z + z , t + t' + 2 Im z • z1) where, z • ~z = £"=1 zfzj . For

u-(z,t)eMn we define \u\ = max.(|Rez;.|, |Imz-|, \t\x/2) where, z = (Zj).

The norm | • | is homogeneous of degree 1 under the one-parameter group

of dilations Dr, r > 0, where Dr(z, t) = (rz, r t). A Heisenberg group left

invariant ball is a set Q of the form

Q - {u e B." : \Uq u\ < 3} for some u0 e B."

and some S > 0. The condition \Uq u\ < ó is equivalent to the pair of

conditions max(|Rez -Rez |, |Imz-Imz|) < ô and |í-í0 + 2 Im z-z0| <

ô , where u = (z, t), u0 = (zQ, t0). u0 is the center of the ball Q and <5 > 0

is the radius. H" balls are tilted (2n+1 )-dimensional rectangles, with maximum

tilt from the center of Q, proportional to the radius of Q times the Euclidean

distance of the projection of the center of Q onto C" , to the origin.

An H" atom is a function a(z, t) supported in a ball Q and satisfying

fa(z, t) dzdPz dt = 0 and \a\ < \Q\~xXq ■ (We denote by XB the character-

istic function of a set 5 .)

Let L(z) be a homogeneous distribution on C" of degree -In , which agrees

with a smooth function away from the origin. Also let ô(t) be the Dirac dis-

tribution in the t variable. We denote by K the distribution on H" defined

by K(z, t) - L(z)ô(t). Thus, if 0 is a test function and (K, <j>) denotes the

action of K on <f>, then (K, 4>) = (L, </>(• , 0)). Geller and Stein [GS1, GS2]

Received by the editors August 23, 1989 and, in revised form, November 14, 1989.

1980 Mathematics Subject Classification (1985 Revision). Primary 43A80.

©1991 American Mathematical Society0002-9947/91 $1.00+ $.25 per page

435

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

436 LOUKAS GRAFAKOS

proved that the operator A : C™ -» C°°(Mn) given by Af - f*K extends to a

bounded operator from Lp to Lp for 1 < p < oc . ( * denotes the group con-

volution.) They established their estimates by embedding the operator A in an

analytic family of operators Ay such that A = A and by proving an L and

an Lp estimate at the endlines of a strip containing 0. Their result followed

by analytic interpolation. It has been an open question whether the operator

A is of weak type (1,1). In this paper we establish a weaker estimate, still

sharper than the Lp boundedness of A, namely we prove that A is of weak

type Hx. By this we mean that A extends to an operator that maps HX(U")

to weak Lx(Mn). Here Hx(B.n), henceforth Hx, denotes the Heisenberg group

Hardy space, homogeneous under the family of dilations Dr, r > 0, defined

as follows: Hx is the set of all sums of the form / = Yl ^naQ > where the ß's

are Heisenberg group balls, XQ e C, £ \XQ\ < +00, and the aß's are atoms

supported in the corresponding balls Q. \\f\\Hi is then defined to be the infi-

mum of Y \^n\ over au representations of / as J2 XQaQ . Our main result is

the following:

Theorem. The operator A, as defined before, is of weak type Hx.

The proof is an application of the method developed in [C]. Some technical

difficulties arise because of the geometry of the Heisenberg balls.

2. Preliminaries for the proof

For ct,t€Z, T>fT,we define the class Ra T of all sets of the form

B = $z, i)el" :max(|Rez.-Rez°|, |Imz,. - Imz°|) < Ia( j J J J J

1for all (zQ, i0) € H" . For t = a , Ra a is the class of all Mn balls of radius 2a .

Given B e Ra r let a(5) = a, t(5) = t denote the associated parameters.

The corresponding Euclidean rectangle 5Eucl of 5 e Ra T is the set

5Euci = i(z,t)eMn: max(|Re z. - Re z)\, |Im z. - Im z)$ < Ia

and |/ - /0| < 2<7+T

where (z0, tQ) is the center of 5 . Let Ra T denote the class of all sets in Ra

whose corresponding Euclidean rectangles have vertices closest to the origin of

the form

(ix2a, ... , in2°, jx2a, ... ,jn2a, k2a+T),

where /,,... , in, jx, ... ,j„,k are integers.

and \t - t0 + 2 Im z • z0| < 2o+x


WEAK TYPE ESTIMATES ON THE HEISENBERG GROUP 437

For 5 e Ra x, 5* will denote an expansion of 5 by a constant factor that

has the following two properties:

(a) If B' e Rai xi, a < a, x < x, and B' intersects 5 then B' ç B*.

(b) If D e Ra> zi, a < a, x < x, and the projection of D onto C" is

contained in the projection of 5 onto C" , then D* ç 5*.

Note that any larger expansion of 5* (for example 5**, 5***) also satisfies

(a) and (b).

Given U, V subsets of H" , we denote by U • V the set of all uv where

u e U, v e V, and we denote by U~ the set of all u~ where u e U.

Following Christ [C], we define the tendril T(q) of a set q e \Ja \Jx>a Ra T

to be the set T(q) = q* ■ {(z, 0) : \z\ < 2T(i)+1}. One can easily check that

\T(q)\ ~ 2a(q)+(2n+X)T(q). (We denote by \B\ the Lebesgue measure of the set

5.)

For the proof of our theorem, we may assume that a given / e Hx is a finite

sum J2 ôaQ ' where J2 \^n\ - 2||./]|#i • Once the theorem is proved for such

/ the general case will follow by a limiting argument. We may also assume that

each Xq in the decomposition of / is positive since we can always multiply an

atom by a scalar of modulus one to achieve this. Finally, we can assume that

for any ball Q in the decomposition of / we have a(Q) e Z. The general

case follows from the observation that every ball Q is contained in a ball Q'

with comparable measure and with a(Q') e Z. Let us call !? the collection of

balls appearing in the atomic decomposition of / e H .

We are now given an a > 0 and a finite collection !? of balls Q with

a(Q) e Z and with associated scalars XQ > 0. The first of the following

two lemmas is valid in any space of homogeneous type X. For our purposes

X = M" .

Lemma 1. Given a > 0 and ^ as above, there exists a collection of balls {S}such that:

The balls obtained by shrinking the S's by a fixed constant factor

(1.1) are pairwise disjoint and no point in X is contained in more than

M S's. M is a constant depending only on X.

;i.2) ^Tisi^CcT'^^.S Q€F

;i.3) £¿0<c4S|.QCS"

11-4)Q<£ any S'

< Co,

L°°


438 LOUKAS GRAFAKOS

Proof. Let F = Ee6jr*clßl %• and let

Q = { x e X : sup |5f ' f F >aB3x Jb

B balls

Í2 is an open set. Then by [CW, Theorem (1.3), p. 70], there exists a sequence

of balls Sj such that Q = |J. S,, a fixed dilate of every S. meets the com-

plement of Í2, and such that (1.1) is satisfied. (The first statement in (1.1) is

not explicitly mentioned in the statement of Theorem (1.3) in [CW] but fol-

lows easily from the proof.) Using the fact that the Hardy-Littlewood maximal

function in a space of homogeneous type is of weak type (1,1) [CW, Theorem

(2.1), p. 71] and with the help of (1.1) we immediately derive (1.2). We only

need to prove (1.3) and (1.4). For fixed j we have

E h = i E W'*e ^ cî \ST LF* CaW-QQS; J QCS] JS1

The last inequality is valid because a fixed dilate of S. meets the complement

of Q. To check (1.4) fix x0 e X. Let Q0 be a ball with the smallest possible

radius that contains x0 and is not contained in any S. . Then

E *Q\Q\~lXQ(x0)< E >-q\Q\~1 XQAx0)ejfanyS; ß£anyS;

= \Qo\~l f E ¿flißfVîGor1 / F-

Let Sq be any S containing x0. If Q0 had smaller radius than S0 then

Q0 Q Sq , which is impossible. Thus S0 has no larger radius than QQ which

implies that S0 ç Q*0 . Therefore some dilate of Q*0 , call it g* > raeets the

complement of Í1. We then have \Q*Q\~X fQ*F < a. Finally |ö0|_1 fQ F

< C|ß0|~ Jq* F < Ca and thus (1.4) holds. This concludes the proof of

Lemma 1.

Call W the subcollection of y consisting of all balls Q contained in S*

for some S as in Lemma 1.

Lemma 2. Given an a > 0, and & and fê as above we can find a measurable

set E CEn and a function k:W—>Z such that

(2.1) \E\<Ca~X £Ae,

(2.2) ß-{(z,0):|z| <2J+X}CE for all Q eW and all j <k(Q) ,

(2.3) ifQ c S* then k(Q) > o(S*) > o(Q),



(2.4) £ XQ<22n+Xa22n+\ ~o+{2n+l)x

k(Q)<x

for any a, x e Z, x > a, and any q G 5

(Here 5* denotes any expansion of B G (JCT UT>CT ^v T âi satisfies (a) a«i/

(b).) The constant C depends on the dimension n and not on a, &~, {XQ} .

Proof. Find a t0 > a(Q) for all ß e & such that Ee€g?¿e < a22("+2)T°.

Initially set ^ = 0 and %?2 = 0. Also set Tj = t0 - 1, a¡ = rx. Select all

«(Tîln+l)!,XQ>a2°^

QCq"

q £ Rn . such that

2)

Any ß contained in q* for some # selected is assigned to q and is placed in

^x. Notice that any ß of class 9^ is assigned to at most K many <? selected,

where K is a constant depending only on the definition of q*. This was step

(<7[, t,) of the induction. Now set a2 = ax - 1. Select all q e Ra T such that

E Ae>a2(T2+(2'!+1)T'.

Qcg'C69\(*;u«£)

Any ß contained in g* for some # selected is assigned to <? and is placed

in Wx . This was step (a2,xx) of the induction. Continue similarly until we

reach a a smaller than minQ6g,cr(ß). Then place in 9^ all ß with a(Q) = t,

which are not already in ^ .

Now set t2 = Tj - 1, a2 = t2 . Select all q e Ra T such that

E Ae>a2'72+(2"+1)T2

ce«*

Any ß contained in <?* for some q selected is assigned to q and is placed in

Wx . We just described step (a2, x2) of the induction. Repeat this procedure

until all the a's are exhausted and then place in W2 all ß € S? with a(Q) = x2

which are not already in Wx . Continue the double induction until we reach a

T<minß6rrj(ß).

We have now split W into two disjoint classes Wx and W2. For ß e ?,

define k(Q) - max s(l + x(q), 1 + a(S*)), where the maximum is taken over

all q such that ß is assigned to q and over all S such that ß c S*. For

ß € ?2 define «r(ß) = max5(l + a(S*)), where the maximum is taken over all

S such that ß c S*. (2.3) is now clearly satisfied. To simplify our notation,

for q e U<r UT>CT Ba t set A(fl) = Eqc?* ê ' where the sum is taken over all


440 LOUKAS GRAFAKOS

Q e ff not yet placed in ffxuW2 at step (a(q), x(q)). To prove (2.4) fix

q G Ra T and consider two cases. If q was not selected then

,<t+(2h+1)ta2<T+(2"+1,T>A(i)> E V

QCq"K(Q)<X

Suppose now that q was selected at the step (a, t) of the induction. Let us

assume that (2.4) fails, i.e.,

E, ^ ~2rt+l -CT+(2n+l)îXQ>2 a 2

QCq'K(Q)<X

Note that for (q1)* D q*,

Mq')>Mq)> E VQCq'

K(Q)<X

It follows that, at least when x < t, , the unique q G Ra x which contains

q would have been selected at step (a, x + 1) ; a contradiction because all

ß G & contained in q* would be contained in (q1)* and therefore they would

have been placed in %?x at a previous step. When x = xx choose q e Ra+X

such that q'EucX D qEucX. Then by (b), q* c (<?')* and the previous argument

leads to a contradiction. Define E - M selected r(?) U |J5 5'+ , where for any S,

S+ = S*- {(z, 0) : \z\ < 2a{S')+2}. Since

U*+<Ei5+i^cEi^Ca EV5 S Q6^"

to prove (2.1) it will suffice to bound | LL selected T(q)\ ■ We have

£ |7Xí)|<C E 2ff(<?)+(2'î+1)T(î)

? selected ? selected

^Cq_1 E Am^-1 E E *fl? selected g selected Q assigned to q

< Col E ^ß *^ : ß *s assigne(it0 ^}

</s:Ca ' E ac<Cq_1 E aQ-Q€^, QeF

Finally, we check (2.2). If ß G 9¡ and ; < fc(ß) then

ß • {(z, 0) : |z| < 2J+1} Cq.{(z,0): \z\ < 2x(q)+X} = T(q) ç E ,

where ß is assigned to # . If ß G 9^ , let S0 have the largest radius among all

5 such that ß c S*. Then k(Q) = 1 + a(S*) and for ;' < k{Q)



ß.{(z,0):|z|<2J'+1}c.S0+c£.

The lemma is now proved.

3. Proof of the theorem

We must show that

(i) \{uemn\(Af)(u)\>a}\<Ç-\\f\\Hl.

We split / as g + b, where

b = E Vö ' 8 = E Ve ■Qew Qeyw

By (1.4), ||g||L°c < Ca and thus

C

Q^\^ a Qe&~g\\¡} < Ca \\g\\Ll <Ca E XQ-ä 2 X°- ~ a "

The L2 boundedness of A in [GS2] gives

\{u G H" : \(Ag)(u)\ > f } < 4 H^lli2 $ -2 UWÍ < % ll/V ■ii z j a a a

Therefore ( 1 ) will follow from

(2) \{u£M":\(Ab)(u)\>^}\<^\\f\\H>.

Because of (2.1) and of the assumption Egeô - ^ll/Hî, (2) will be a

corollary of

\{ueUn\E:\(Ab)(u)\>^}\<^ EACß€iT

which in turn will follow from

(3) M*l&(„»\*) < Ca E ê

with the aid of Chebychev's inequality.

The rest of the paper is devoted to the proof of (3).

Fix <f> g C0°°(C") supported in 1/2 < \z\ < 2 such that

E 0(2 Jz) = 1 for all z G Cn - {0}.jez

We decompose the operator A = Ey ¿j » where A- is given by convolution

with Kj(z, t) = L(z)<p(2~J z)ô(t). Since all balls ß appearing in the definition

of b are in W, in the remainder of the paper every Q considered is in &. This

restriction is assumed to hold in all sums below.


442 LOUKAS GRAFAKOS

To treat (3) fix m G B."\E and write

Ab(u) = A (E Vß) (")

= E((Eve)**,>)E(ve* E *,)oo

EEi>0 ;'6Z

E Ve**/»(fiiw-j

(«).

If we set

Bk= E Vß> ̂ eK(ß)=*

(3) will be a consequence of

|2

(4) YAbj-.*ki)j&L

< Ca2 s E xq for all s > 0 .

L2(H")K(ß)=;-i

Without loss of generality, we may assume that K(z, t) = L(z)ô(t) is a real-

valued distribution. Expanding the square out, we write the left-hand side of

(4) as:

(5) E\\\Bj-s*Kj\\l> + 2 E [(Bj^KjXB^tKJdzdJdt

+ 2 E ¡(Bj^KjXB^tKJdzdZdtk<j-3

We will treat the first and the third term inside the brackets in (5). The estimate

for the second term in (5) will follow from the estimate for the first term with

the use of the Cauchy-Schwarz inequality. We therefore only need to prove

(6) j€Z

Bj-s*Kjfû+ E jiB^KfiB^KJdzdldtfc</-3

<Ca2's E V

K(Q)=j-S

By rescaling, it suffices to prove (6) when j = 0. Our proof will be complete if

we can show:

(7) ||5_i*/^0||22<Ca2-i E kQ andK(Q) = -S

¿2\[(B_s*K0)(Bk_s*Kk)dzdJdt <Ca2~s £ XQ.(8)k<-3 K(Q) = -S



We start with the proof of (7). We will need two lemmas.

Lemma 3. For any ball Q with a(Q) < 0, \\aQ * K0\\L«> < C2_<T(e).

Proof. Fix a ß with a(Q) = a < 0. We have

\(ao*K0)(z, t)\ = \ I a0(z -w,t-2 Imz -w)LJw) dw dwu \Jc"

<C\Q\-X\SQ(z,t)\ = C2-{2n+2)a\SQ(z,t)\ ,

where

SQ(z, t) = {w G C" : \ < \w\ < 2 such that (z - w , t - 2 Im z • w) G ß}.

Clearly

SQ(z, t) = {w g C" : \ < \w\ < 2 such that (w, 0) G ß~'(z> 0}-

(We set Q'x(z, t) = Q~x ■ {(z, r)}.) Since ß is a left-invariant ball, ß0 =

Q~x (z, t) is a right-invariant ball of radius 2° , centered say at (zx, tx). The set

Sq(z , t) is empty unless \zx | ~ 1. This observation implies that the maximum

tilt of ß0 from its center is < C2CT . Thus SQ(z, t) is contained in

{w€Cn :\w-zx\ <2a, |2Imz1-üJ| <C22a}.

Setting £ = w - zx, to prove the lemma it will suffice to show that

|{CgC": |C|<2ct and|Im(C-z,)| < C22ff}| <C2(2n+3)CT.

The latter becomes obvious when we introduce a rotation that takes zx to the

point (a + iO, 0 + i0, ... ,0 + i0) eC" , where \a\ < C.

Given f(z, t), a function on the Heisenberg group, let / denote the function

f(z, t) = /((z, t)~ ) = f(-z, -t). Note that for all /, g, h, functions on

any group with Haar measure dx, the following is valid:

J(f*g)hdx = I f(h*g)dx.

To prove (7) we will need one more lemma.

Lemma 4. KQ * KQ has compact support and satisfies

\(K0 * K0)(z , 01 < C(|z| + III)-1, \V(K0 * K0)(z, f)| < C(|z| + |i|r2 .

Proof. Let y/ be a test function. Then

(K0 *K0,y/) = (K0,y/*K0)= / L0(u;)(^ * a:o)(ií; , 0) dw dwJc"

(9) =/ LJw) [ V(w-Ç, -2Imw-X)L0(Odi;dXdwdw

= [ [ L0(z + Oy,(z,-2lmz-OL0(Ç)dÇdÇdzd-z.Je" Je"


444 LOUKAS GRAFAKOS

Let C = (d , C2, • • • . C„), Cj = Xj + iyj, 1 < j < n . Fix / : 1 < / < n and

denote by z , Ç' G C"_1 the points z, Ç whose /th variable is deleted. Change

variables in (9) by setting

t = 2 Im z ■ C = -2 Im z^ - 2 Im z • Ç

and

s = -2 Re z • Ç = -2 Re z¡C¡ - 2 Re z • £ .

Then Cfc = (5 - « + 2z^ • C')/ - 2z, and |det(Ö(^, y,)/d(s, 0)1 = il^/f2 •We can rewrite (9) as

(10) / W(z,t)\\\z,\-2 f f L0(z + C)L0(Qdsdi:'dC'JC"xR Jc"~' JR

s - it + 2i ■ C'

dzdz dt,

where

c- c,.c 1 ' -2z,. c.1+1 ' .c.

It follows that (K0 * K0)(z, t) is equal to the expression inside the brackets in

(10). Since L0 is supported in |z| < 2 we must have

5 - it + 2~z ■ £ I(11) \z +CI + z,+

-2z,<C

which implies | - 2 \z¡\ + s + 2 Re ~z • £\ < C\z¡\. Therefore, once C' is fixed,

s varies over a set of measure < C\z¡\. It follows that

¡(K^K^z^^Kjlzf2 [ [\\L0\\2L~dsdC'<C\:,-1

i-lSince / was arbitrary in {1,2,...,«} we get \(K0 * K0)(z, 01 < C|z| . It

also follows from (11) that \t - 2 Im ~z ■ Ç'\ < C\zk\ which implies that

|i|<C|zfc| + 2|z'||C/|<C(|z,| + |z'|)<C|z|.

Thus on the support of (K0 * K0)(z, t), \t\ < c\z\ and the asserted bound

follows. The estimate for the gradient can be proved similarly.

We now begin the proof of (7).

First fix 5 > 0 and assign each ß occurring in (7) to some (but only one)

element q G 5CT(e) 0 suchthat ß intersects q (hence ß c q* ). For q in Ra 0

(a < 0) write

Aq= E ^ßflß ' ^q = E ^Q-Q assigned to q Q assigned to q

Also split every q G Ra 0 (a < -s) in 2s, q¡ £ Ra _s, 1 < ; < 2s, so that

their union is q and associate with each ß assigned to q one q. such that ß

intersects q. (hence ß c q* ). Write

Eg associated to a

XQaQ' £ A,ß associated to í .



We then have An = eLi An > ̂ = E,«i K ■ We now write the left-hand side

of (7) as

E E V*o

= 2EE E E [(Aq*KQ)(Aq,,KQ)dzdJdta'<-s a<a' ?'€*„< 0 «e^.o

<I + II,

where

1 = 2 E E E E \f(Aq*K0)(Ag,*K0)dzdz-dta'<-s a<a'q'€Ra, 0 9€Rc0

qn(q')'¿0

11 = 2 E E E E |/(v*o)K'**o)^^¿'<t'<-í<t<(t' (?'€*,< 0 <?€*„,<,

«n(í')*=0

To treat term I, let us fix a < -s, a <a , q e Ra, 0 . Then

E J(Aq*KQ)(Aq,*K0)dzdJdt

(12)< E K'*ôIIl'K*ôIIl» (by Lemma 3)

?n(í')V<3

< e CV2~*V»n(9')V0

To get a bound for (12) write (a1)* n (vertical Euclidean cylinder over #') as a

union of C 2s, q} e Ra _s. We then have that U,(<7/)* 2 (<7)* and therefore

C2S C2S C2S

E Ê E ¿Ê E *,*£ E Vq€R.,0 j=\ q€R,0 j=\ ?£*„,„ j=\ ÖC(«J)*"

«n(«')V0 ?n(?J')V0 8Ç(«y')** f(ß)=-i

An application of (2.4) for each </. gives that the above is bounded by

,-,~S,,n+l r,0-(2n+\)S . ,-, «lT-2«5Cz z az < Caz

Therefore (12) is bounded above by CXq,2~°Ca2a~2ns < CaXq,2~s.

Summing over all q G Ra, 0, a < a', and a' < s we get the desired

conclusion for I.

In the sequel we will need the following simple lemma whose proof we omit.


446 LOUKAS GRAFAKOS

Lemma 5. On the Heisenberg group, let a have support contained in the set A

and integral 0 and let h be C . Then

\(a * h)(u)\ < \\a\\L¡ \\Vh\\L°c,A-i.u) Euclidean diameter of(A~ • u).

To treat term II, fix as before a < -s, a < a', q G Ra* 0. Note that for

q G Ra 0 and all u G H" , the Euclidean diameter of q~xu is at most C2~s.

We have

qn(q')"=0

E \[(Aq*K0)(Aq,*K0)dzdldt.7,0

r=0

< E \f(A,*K0){Aq.*K0)dzdzdt

m{q')'=0

< E \JAq(Aql*K0*K0)dzdzdt

qn(q')m=0

^ E Xq'WAq*Ko*Koh~(q') (by Lemma 5)

qn(q')'=0

(13) <C2~sXq, E ¿gllv(*o**o)llL-(i-vr

qn(q')'=<S

To get an estimate for (13) consider all expansions of q by a factor of 2m,

m > 0. By this we mean all sets q'm in 5m+(J- 0 with the same center as q

and tilted as q . We first fix an m > 0 and get an estimate for

£ V«n?;+l^0

Note that since KQ*K0 has compact support only those m for which m+a < C

are of interest to us. Consider two cases:

Case 1. m + a < 0. Write q'm+x as the union of 2~(m+(7 ' rectangles q'm+x k G

Rm+n' «W > ! < k < 2m+"' ■ Then

{m+ff ) ^ —(m+ff )

E M E E V^ £ E ¿</€«,,_, /c=l 9€/C _s fc=l Cc(Cl.t)**

?n?;+1^0 qnq'm+lk¿0

. ~—(m+o')s~, ~m+o'+(2n+\)(m+a') ^ «(2n+l)(m+<;'< Z Caz = CaZ

where in the last inequality we used (2.4).



Case 2. 0 < m + a < C. All q G Rg _s which intersect q'm+x intersect at

most C rectangles Rk in 50 0. Then

E Ê E K^t E ^c«-q£Ra,-s *-l <i£R,,-s k=\ QCR'k'

qnq'm+¡¿0 qnRk¿0 RkeR0,o

We finally get an estimate for (13):

C-a'

(13) < C2~\ E E K Hvô*ô)IIl-(?-'V)m=0 qeR

C-a'

a, —s

qn(q'm+l-q'm)¿0

<cr\'E E Vm=0 ?€/?„_,

-2(m+cr

«n9',#0

<C2 sXq,£2

Lm=0

-2{m+a )(-, ~{2n+\)(m+oC-a1

+ £ 2"m=—<r'+l

■2(m+<r )Ca

< Ca2 *Xq,

Summing over all q G R^ 0, all a < a , and all a' < -s we get the desired

conclusion for term II. This concludes the proof of (7).

4. The off-diagonal terms

This section is devoted to the proof of (8). We begin with a lemma.

Lemma 6. For k < -3, Kk* K0 is supported in the set {(z, t) : \z\ < C, \t\ <

C2 } and satisfies

(6.1)

(6.2)

(6.3)Proof. The same reasoning as in Lemma 4 shows that, when I — n ,

(Kk*K0)(z,t) = j\zn\-2 [ [ Lk(z + C)L0(QdsdC'dt* Je Jr

where £' = (£,,... ,i JeC""1 and

-k

-2k

11^*^0^-<C2 \

||V(^*^0)||Lco<C2

||V2(^*^0)||LOo<C2-3fc,

n-\

S-U + 2Z -C''

-2z„

Since Lk(z) is supported in \z\ ~ 2 we get that

s-it + 2-z • f'|z +CI + Z* +-2z.

<C2 ;


448 LOUKAS GRAFAKOS

thus C' integrates over a Euclidean ball about -z of radius C2 . The above

inequality gives

|j-2|zH|2 + 2Rez'-C'|<C2A:|z„l,

|/-2Imz'-C'|<C2':|z¡fc,

n>

We certainly have that \zn\ < C on the support of (Kk * K0)(z, t). It follows

that for any fixed C', s integrates over a set of measure < C 2 . Also

|i|<C2*|zJ + |2Imz'-Ç/|

<C2/: + 2|Im(z, + cVc'|

<C2* + 2|z' + i'|K'l <C2*.

This proves the assertion about the support of Kk * K0 .

We now prove the size estimates for Kk* KQ. It follows from the definition

of C that the plane zn - 0 does not intersect the support of Kk * K0 . Since

the latter set is compact, there exists a constant C0 such that \zn\ > C0 when

(z, 0 lies in the support of Kk * K0. We use ||I/fc||Loo < C(2 ) to estimate

II^*ôIIl~ by

-7 Cr, / / \\Lk|Loo||Ln|Loo dsd(' dX4 ° J\r'+z'\<C2k J\s-ac,\<C2kJ\s-ac,\<C2k kL °L

< C(2k)2{n~x)2k(2k)~2n = C2~k ,

where in the above estimate we set ar> - 2\zn\ - 2 Re ~z • £'. The estimates

for the derivatives of Kk * K0 are similar.

For each k < -3, let us call Sk = {(z, t): \z\ < C, |i| < C2k} so that

support (Kk *K0) çSk. Note that for each u G M" we have Skxu = Sku.We will need the following two lemmas.

Lemma 7. Fix any Q e W with a(Q) = a and any k < -3. If m = max(a, k)

we have

(7.1) E XQ,<Ca2m.

K(Q')<m

Lemma 8. For any ßef with a(Q) = a, any k <a, and any (w , s) e B." ,

(8.1) j xSk(zJ){w,s)dzdz-dt<C2k+(2n+X)°.

Proof. For any ueQ, Smu is a tilted rectangle in R2"+1 of dimensions

(C,... ,C ,C2m).s-^-'

2n times



It follows that \JueoSmu is contained in a tilted rectangle in R2n+X of dimen-

sions(C... ,C ,C2m).

-v-'

2n times

Cover \Jn&QSmu by the union of c2"(2"+1)m elements of Rm m . For each

element of Rm m apply (2.4). The desired conclusion follows.

To prove (8.1) note that the set of all (z, t) e Q for which the fixed (w , s)

lies in Sk(z, t) is contained in a rectangle 5 of dimensions

(C,... ,C ,C2h)

2n times

centered at the center of ß and with maximum tilt from its center ~ Cd,

where d denotes the Euclidean distance from the center of ß to the origin, ß

has dimensions(2CT,...,2CT ,22a)

2n times

and has maximum tilt from its center ~ C2° d (a < 0). It follows that ß and

5 intersect almost vertically and with respect to the coordinate system induced

by R, Q should be thought of as having dimensions (2°, ... ,22a, ... ,2a).

Therefore ß n 5 has measure at most

f, ~.a -cr -2<T-min(<:,<j) _ ^,~k+(2n+l)a

-'—J-^-' '2«-l times

We can now prove (8). Write the left-hand side of (8) as:

E f(B_s)(Bk_s*Kk*K0)dzd^dt

^ E E h E \[(AQ)(Bk_s*Kk*K0)dzd^dta<-s K(Q)=-s k<-3 '

a(Q)=a

< I + II,

/fc<-3

where

!=E E EÀQÎ\AQ\\Bk-s*Kk*Ko\dzdz-dt,a<-s k(Q)=-s k<a

a(Q)=a

11 = E E E h I [(AQ)(Bk-s * Kk * *o)dz dz dt.a<-s k(Q)=-s k<a 'J

a(Q)=a

We begin with term I. Fix a < -s and ge? with k(Q) = -s and a(Q) = a .


450 LOUKAS GRAFAKOS

By Lemma 5 and the definition of an atom we get

E¿e/i«k<a J

Q\\Bk_s*Kk*K0\dzdzdt

< edgier1 E /o(Q')

E V2 °(Q)\MKk*k{0/\\L°

K(Q')=k-s

[Q'nsk-¡(z,t)¿0

> dzdldt

(by (2.3) and (6.2))

<CXQ\Q\ 'E2* *2 2k f E ï-çydzdzdtk<a Q K(0')=k-sK(Q)=k-s

Q'r\Sk{z,t)¿0

< CX0\Q-1 y 2-k-s Í

ta JQ I E Aß'lö'l XQ'{w,s)Q'nsk{z,t)jt0

I K(Q')=k-S

Xs (z,t)(w> J) dwdw ds dzdz dt

<CXQ\Q\-Xl£2-k-s[ E ¿ß'lö'fV«'.*)

e'nlUQ(V)¿0K(ß')=A:-J

(by Fubini)

fc<CT

/XSk{zJ){w,s)dzdzdt dw dw ds

(by (8.1))

< CXQ\Q\~X E 2~k~s 2k+{2n+X)a E Aß'

k<" ß'n(Uueß^«)^0

K(ß')=A:-i

s^'T E V

K(ß')=^-5

<CAe2-í-ff E V (by (7.1))

e'n(lU5»")*0*(ß')<<7

<Ca2~X-



Summing over all ß G f with a(Q) = a , /c(ß) = -5, and all a < -s we get

the required conclusion for term I.

To treat term II, fix a < -s and ß G W with a(Q) = a and k(Q) = -s.

Two applications of Lemma 5 give the first two inequalities below.

E^ßl/k>a ]J

AQ(Bk_s*Kk*K0)dzdldt

<CyEAQ2a\\Bk_s*V(Kk*K0)\\L~IQ)k>a

< C EVûp E V'^^H **o)«L-k>a ueQ Q'r\Sklu¿2¡

K(Q')=k-s

(by (2.3) and (6.3))

<CEVCT2^2-3fcsup E V

k>a ueQ ß'n(lJueoS,«)^0

K{Q')=k-s

<cxQ2-s¿22°-2k e V

k>° e'n(U„ee5*")^0K(Q')<k

< CXQ 2~s E 2a~2k(Ca2k) < CaXQ 2~s.

k>a

The penultimate inequality follows by another application of (7.1). Summing

over all ß G W with rj(ß) = a, tc(Q) - -s, and all a < -s we get the

required conclusion for term II. (8) is now proved. This concludes the proof of

( 1 ) and hence of our theorem.

5. Concluding remarks

The proof follows the method initiated in [C] in the treatment of the max-

imal function along the parabola (t,t ). An application of this method gives

Theorem 2 in [G]. The same method might prove that the analytic family Ay

considered by Geller and Stein in [GS2] maps Hx to Lx'°° when Re y = 0.

This result, together with the sharp L estimates in [GS2], would give a positive

endpoint result for the analytic family A7. It still remains an open question

whether the operator A is of weak type ( 1, 1 ). The answer to this problem

is probably the same as for the Hilbert transform along the parabola (t, t ) in

R2.

I would like to thank my supervisor Mike Christ, for introducing me to this

problem and for giving me numerous suggestions throughout this work.


452 LOUKAS GRAFAKOS

References

[CC] S. Chanclo and M. Christ, Weak type (1,1) bounds for oscillatory integrals, Duke MathJ. 55(1987), 141-155.

[C] M. Christ, Weak type (1,1) bounds for rough operators, Ann. of Math. (2) 128 (1988),19-42.

[CG] M. Christ and D. Geller, Singular integral characterizations of Hardy spaces on homogeneousgroups, Duke Math. J. 51 (1984), 547-598.

[CR] M. Christ and J. L. Rubio de Francia, Weak type (1,1) bounds for rough operators. II,

Invent. Math. 93 (1988), 225-237.

[CW] R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces

homogenes, Springer-Verlag, Berlin-Heidelberg-New York, 1971.

[FS1] G. Folland and E. Stein, Hardy spaces on homogeneous groups, Princeton Univ. Press,

Princeton, N.J., 1982.

[FS2] _, Estimates for the db complex and analysis on the Heisenberg group, Comm. Pure

Appl. Math. 27 (1974), 429-522.

[GS1] D. Geller and E. Stein, Singular convolution operators on the Heisenberg group, Bull. Amer.

Math. Soc. (N.S.) 6 (1982), 99-103.

[GS2] _, Estimates for singular convolution operators on the Heisenberg group, Math. Ann. 267

(1984), 1-15.

[G] L. Grafakos, Endpoint bounds for an analytic family of Hubert transforms, Duke Math. J.(to appear).

[S] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ.

Press, Princeton, N.J., 1970.

Department of Mathematics, Box 2155 Yale Station, Yale University, New Haven,

Connecticut 06520


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WEAK TYPE ESTIMATES FOR A SINGULAR CONVOLUTION … · fa(z, t) dzdPz dt = 0 and \a\ < \Q\~xXq (We denote by XB the character-istic function of a set 5 .) Let L(z) be a homogeneous